SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Using Lemma 5 and Lemma 6, we offer an expansion viewpoint on the vector-valued functional gradients of  $\mathcal{L}$  [41, 16]: Suppose we have a specific example vector  $(\boldsymbol{x}, \boldsymbol{y}) = [(x_{i,j_i}, y_{i,j_i})]^d \in \mathcal{X}^d \times \mathcal{Y}^d$ , the gradient  $\boldsymbol{\mathcal{G}}$  of the multi-learner loss function  $\mathcal{L}$  w.r.t. the vector-valued model  $\boldsymbol{f}$  can be expressed by
$$
\mathcal {G} (\mathcal {L}; \boldsymbol {f}; (\boldsymbol {x}, \boldsymbol {y})) = \left[ \partial \mathcal {L} _ {i} / \partial f _ {i} \right| _ {f _ {i} \left(x _ {i, j _ {i}}\right), y _ {i, j _ {i}}} K _ {x _ {i, j _ {i}}} \bigg ] ^ {d}. \tag {11}
$$
We also broaden the applicability of RFT and GFT from their single-learner versions [73] to a multi-learner one. Under this context, RFT involves randomly picking examples for each learner, while GFT selects examples that satisfy
$$
\left( \right.\boldsymbol {x} ^ {t ^ {*}} = \underset {[ x _ {i} ] ^ {d} \in \mathcal {X} ^ {d}} {\arg \max } \left\| \right.\left[ \right. \partial \mathcal {L} _ {i} / \partial f _ {i} \left. \right| _ {f _ {i} ^ {t} (x _ {i})} \left. \right] ^ {d} \left. \right\| _ {\mathcal {H} ^ {d}}, \quad \boldsymbol {y} ^ {t ^ {*}} = [ y ^ {t ^ {*}} ] ^ {d} = \left[ f _ {i} ^ {*} \left(x _ {i} ^ {t ^ {*}}\right)\right] ^ {d}\left. \right) \tag {12}
$$
To avoid clutter in the notation, our examination is restricted to the selection of a single example for each learner during every iteration, and we provide the pseudo code in Appendix A.
In the upcoming discussion, we shall present our theoretical examination of the convergence performance of multi-learner RFT and GFT. Our approach differs from [73] as we focus on RFT's average performance by introducing the expectation operation over random sampling. This helps us gain valuable insights by averaging out the impact of randomness. Recall the teaching settings (Eq. 6, 9), we then proceed with our analysis of RFT's per-iteration reduction concerning  $\mathcal{L}$ .
Lemma 7. (Sufficient Descent for multi-learner RFT) Suppose there are  $d$  learners, and the example mean for each learner is  $\mu_{i} = \mathbb{E}_{x_{i}\sim \mathbb{P}_{i}(x_{i})}(x_{i}) < \infty$ , and the variance  $\sigma_i^2 = \mathbb{E}_{x_i\sim \mathbb{P}_i(x_i)}(x_i - \mu_i)^2 < \infty, i\in \mathbb{N}_d$ . Under the assumptions outlined in both 3 and 4, if  $\eta_i^t\leq \frac{1}{2L_\mathcal{L}\cdot M_K}$  for all  $i\in \mathbb{N}_d$ , then RFT teachers can, on average, reduce the multi-learner loss  $\mathcal{L}(\pmb {f})$  by:
$$
\mathbb {E} _ {\boldsymbol {x} \sim [ \mathbb {P} _ {i} (x _ {i}) ] ^ {d}} \left[ \mathcal {L} (\boldsymbol {f} ^ {t + 1}) - \mathcal {L} (\boldsymbol {f} ^ {t}) \right] \leq - \frac {\tilde {\eta} ^ {t}}{2} \sum_ {i = 1} ^ {d} \left(m _ {i, t} \left(\mu_ {i}\right) + \frac {m _ {i , t} ^ {\prime \prime} \left(\mu_ {i}\right)}{2} \sigma_ {i} ^ {2}\right), \tag {13}
$$
where  $\tilde{\eta}^t = \min_{i\in \mathbb{N}_d}\eta_i^t$  and  $m_{i,t}(\dot{x})\coloneqq E_{\dot{x}}[(\nabla_f\mathcal{L}_i(f)|_{f = f_i^t})^2 ]$ .
Intuitively,  $m_{i,t}(\mu_i)$  serves as a measure of the gradient's magnitude (loss  $\mathcal{L}_i$  w.r.t.  $f_i^t$ ) for the  $i$ -th learner at the example mean  $\mu_i$  in the  $t$ -th iteration, and as  $f_i^t$  approaches  $f_i^*$ ,  $m_{i,t}(\mu_i)$  becomes increasingly small. According to Lemma 7, as proven in Appendix B, the average reduction of multi-learner loss  $\mathcal{L}$  per iteration is constrained by a negative upper bound. To be more precise, this upper bound is determined by a range of elements, such as the learning rate, the count of learners, the example variance  $\sigma_i^2$  and the gradient of  $\mathcal{L}_i$  at the example mean  $\mu_i$  ( $i \in \mathbb{N}_d$ ), and these elements are independent of each another. When the gradient at each  $\mu_i$  is large, RFT on average can reduce  $\mathcal{L}$  by a significant amount. Meanwhile, the variance also has an impact on the reduction and this is governed by a diminutive constant of  $m_{i,t}''(\mu_i)/2$ .
The least helpful teaching scenario in the  $t$ -th iteration across the  $d$  learners is represented by  $\min_{i\in \mathbb{N}_d}\left(m_{i,t}(\mu_i) + m_{i,t}^{\prime \prime}(\mu_i)\sigma_i^2 /2\right)$ , which has the smallest gradient at the example mean. It is observed from Lemma 7 that the reduction of  $\mathcal{L}$  per iteration is, at a minimum,  $d$  times greater than that of the worst-case scenario  $-\tilde{\eta}^{t} / 2\cdot \min_{i\in \mathbb{N}_{d}}\left(m_{i,t}(\mu_{i}) + m_{i,t}^{\prime \prime}(\mu_{i})\sigma_{i}^{2} / 2\right)$ . In other words, when multi-learner RFT achieves a stationary point in the worst-case scenario, the multi-learner loss  $\mathcal{L}$  reaches convergence as well. This indicates that the convergence rate of multi-learner RFT is at least as fast as in the single-learner worst-case scenario (also faster than repeatedly teaching).
Introducing the expectation operation enables us to eliminate the randomness that arises from random sampling. In contrast to [73], which determines the decrease based on the discrepancy at specific but randomly chosen example  $x^{t}$  (involving randomness), we establish that the decrease on average is determined by the mean and variance. This valuable insight is important to understand the fundamentals of RFT, which is not considered in [73].
Theorem 8. (Convergence for multi-learner RFT) Suppose the vector-valued model for multiple learners is initialized with  $\pmb{f}^{0}\in \mathcal{H}^{d}$  and returns  $\pmb {f}^t\in \mathcal{H}^d$  after  $t$  iterations, we have the upper bound of  $\min_{i\in \mathbb{N}_d}\left(m_{i,t}(\mu_i) + m_{i,t}^{\prime \prime}(\mu_i)\sigma_i^2 /2\right)$  w.r.t.  $t$  ..
$$
\min  _ {i \in \mathbb {N} _ {d}} \left(m _ {i, t - 1} \left(\mu_ {i}\right) + m _ {i, t - 1} ^ {\prime \prime} \left(\mu_ {i}\right) \sigma_ {i} ^ {2} / 2\right) \leq 2 \mathbb {E} _ {\boldsymbol {x} \sim \left[ \mathbb {P} _ {i} \left(x _ {i}\right) \right] ^ {d}} \left[ \mathcal {L} \left(\boldsymbol {f} ^ {0}\right) \right] / (d \dot {\eta} t), \tag {14}
$$
where  $0 < \dot{\eta} = \min_{l\in \{0\} \bigcup \mathbb{N}_{t - 1}}\tilde{\eta}^l\leq 1 / (2L_{\mathcal{L}}\cdot M_K)$ , and given a small constant  $\epsilon >0$  it would take approximately  $\mathcal{O}(2\mathbb{E}_{\boldsymbol {x}\sim [\mathbb{P}_i(x_i)]^d}\left[\mathcal{L}\left(\boldsymbol {f}^0\right) / (d\dot{\eta}\epsilon)\right)$  iterations to reach a stationary point.
The proof for Theorem 8 can be found in Appendix B. Theorem 8 tells that the minimum of the non-negative term within the upper bound in Theorem 7, which is  $\min_{i\in \mathbb{N}_d}\left(m_{i,t}(\mu_i) + m_{i,t}^{\prime \prime}(\mu_i)\sigma_i^2 /2\right)$ , is also upper bounded, and the iterative teaching dimension is  $2\mathbb{E}_{\boldsymbol{x}\sim [\mathbb{P}_i(x_i)]^d}\left[\mathcal{L}(\boldsymbol{f}^0)\right] / (d\eta \epsilon)$ .
In comparison to RFT, GFT achieves a larger reduction in multi-learner loss  $\mathcal{L}$  per iteration, suggesting a faster convergence rate and a lesser number of iterations required to achieve convergence.
Lemma 9. (Sufficient Descent for multi-learner GFT) Under Assumption 3 and 4, if  $\eta_i^t \leq \frac{1}{2L_{\mathcal{L}} \cdot M_K}$  for all  $i \in \mathbb{N}_d$ , the GFT teachers can achieve a greater reduction in the multi-learner loss  $\mathcal{L}$ :
$$
\mathbb {E} _ {\boldsymbol {x} \sim [ \mathbb {P} _ {i} (x _ {i}) ] ^ {d}} \left[ \mathcal {L} (\boldsymbol {f} ^ {t + 1}) - \mathcal {L} (\boldsymbol {f} ^ {t}) \right] \leq - \frac {\tilde {\eta} ^ {t}}{2} \sum_ {i = 1} ^ {d} m _ {i, t} \left(x _ {i} ^ {t *}\right), \tag {15}
$$
where  $\tilde{\eta}^t$  and  $m_{i,t}(\cdot)$  retain their previous meaning.
The proof of the Lemma 9 is presented in Appendix B. GFT selects examples with the steepest gradient, which leads to  $m_{i,t}(x_i^{t*}) \geq (m_{i,t}(\mu_i) + m_{i,t}^{\prime\prime}(\mu_i)\sigma_i^2/2)$  for each learner. Consequently, it can be observed that per-iteration  $\mathcal{L}$  reduction of GFT has a tighter bound compared to RFT. This is due to the fact that GFT uses a greedy approach to select examples that maximizes the norm of difference between the current and target models. This allows the learners to take a larger step forward  $f^*$  in per iteration. The tighter bound provides theoretical evidence supporting the effectiveness of GFT, which is consistent with the findings in the single-learner teaching [73].
Theorem 10. (Convergence for multi-learner GFT) Suppose the vector-valued model for multiple learners is initialized with  $\pmb{f}^0\in \mathcal{H}^d$  and returns  $\pmb {f}^t\in \mathcal{H}^d$  after  $t$  iterations, we have the upper bound of  $\min_{i\in \mathbb{N}_d}m_{i,t}(x_i^{t*})$  w.r.t.  $t$ :
$$
\min  _ {i \in \mathbb {N} _ {d}} m _ {i, t - 1} \left(x _ {i} ^ {t - 1 ^ {*}}\right) \leq \frac {2}{d \dot {\eta} t} \mathbb {E} _ {\boldsymbol {x} \sim [ \mathbb {P} _ {i} \left(x _ {i}\right) ] ^ {d}} \left[ \mathcal {L} \left(\boldsymbol {f} ^ {0}\right) \right] + \frac {1}{d t} \sum_ {l = 0} ^ {t - 1} \sum_ {i = 1} ^ {d} \left(\| x _ {i} ^ {l ^ {*}} - \mu_ {i} \| _ {2}\right), \tag {16}
$$
where  $\dot{\eta}$  has the same definition as before.
It follows from Lemma 7 and 9 that when  $x_{i}^{t*}$  is close to  $\mu_{i}$  for  $i \in \mathbb{N}_{d}$ , then GFT and RFT perform similarly. In Theorem 10 (The proof is given in Appendix B), we theoretically show this relation by introducing the distance between  $x_{i}^{t*}$  and  $\mu_{i}$ , which provides a deep insight of the difference between RFT and GFT that is not considered in [73]. Specifically, the per-iteration loss reduction under both RFT and GFT has negative upper bounds, and the difference between these two upper bounds can be seen by comparing Theorem 8 and Theorem 10. From a qualitative perspective, GFT can achieve better convergence speed-up because its negative upper bound can take smaller values than that of RFT. This gap is characterized by  $\frac{1}{dt}\sum_{l=0}^{t-1}\sum_{i=1}^{d}\left(\|x_{i}^{l*}-\mu_{i}\|_{2}\right)$  which is the cumulative distance between select  $x_{i}^{t*}$  and mean  $\mu_{i}$  for all learners and averaged over iterations. We emphasize that the purpose of our results is to show the difference between RFT and GFT, rather than proving that GFT always achieves better convergence than RFT (which is not always true). By comparing Theorem 8 and Theorem 10, we can learn that it is possible for GFT to have larger per-iteration loss reduction than RFT. However, we also recognize the intrinsic difficulty to show the exact conditions such that GFT can always be better than RFT. In contrast to our results, the parametric case (e.g., [36]) also has not obtained the necessary and sufficient conditions for greedy teaching to be better than random teaching. More generally, [38] also considers some alternative teaching strategies other than the greedy teaching, such as the parameterized teaching with a multi-iteration reward function. Despite not being able to fully characterize the difference of convergence rate between GFT and RFT, our existing theoretical analysis still poses an important open problem: when and how can GFT provably achieve faster convergence than RFT?
(a) Single-learner teaching
(b) Vanilla MINT
(c) Single-learner teaching
Figure 2: Comparison between single-learner teaching and MINT. (a) Repeatedly invoking single-learner GFT: teaching a white tiger at first and subsequently teaching a cheetah. (b) Simultaneous teaching of a white tiger and a cheetah by GFT. (c) Single-learner teaching of the lion. (d) Partitioning a single lion image into 16 pieces and teaching them concurrently.
(a) Tiger & cheetah.
Figure 3: Comparison of convergence performance between single-learner teaching and MINT. (a) is corresponding to (a)-(b) in Figure 2. (b) is for (c)-(d) in Figure 2. (c) pertains to teaching of a colored lion.
(b) Gray lion.
(c) RGB lion.
4.3 Communicated multi-learner teaching
An infant would often compose previously learnt knowledge in order to grasp a new target concept, such as understanding what a zebra is by combining the learnt ideas of horses and black-and-white stripes. Such an efficient learning motivates us to explore the idea of communicated MINT, which enables the communication between learners. In other words, multiple learners can execute linear combination on the currently learnt functions of all learners [21, 23, 77, 12], that is,  $A^t$  is no longer constrained to be an identity matrix.
In practice, to direct this communication, the teacher can utilize a two-layer perceptron (MLP) to derive the matrix  $A^t$  in Eq. 8 by searching a matrix  $A$  that minimizes  $\| A\pmb{f}^t - \pmb{f}^* \|_{\mathcal{H}^d}$  as much as possible, which is an addition step beyond example selection in each iteration.
Proposition 11. If the proximity between  $\pmb{f}^t$  and  $\pmb{f}^{*}$  is sufficiently close, meaning that  $\| \pmb {f}^t -\pmb {f}^*\|_{\mathcal{H}^d}\leq \epsilon$  where  $\epsilon$  is a tiny positive constant, then  $A^t$  equals the identity matrix  $I_{d}$ .
The proof of Prop.11 is given in Appendix B. This suggests that there is no need for MLP to be used in solving matrix  $A^t$  in every iteration, but only at the beginning, because as the iterations progress,  $f^t$  will approach near to  $f^*$ .
Lemma 12. Under Assumption 3, the communication across learners will result in a reduction of the multi-learner convex loss  $\mathcal{L}$  by  $0\leq \mathcal{L}(\pmb {f}^t) - \mathcal{L}(A^t\pmb {f}^t)\leq 2L_{\mathcal{L}}\| \pmb {f}^t -\pmb {f}^*\|_{\mathcal{H}^d}$ .
Proof of Lemma 12 is given in Appendix B. The difference in  $\mathcal{L}$  between the case where the communication exists and that where it doesn't is lower bounded by zero and upper bounded by the distance between  $\pmb{f}^t$  and  $\pmb{f}^{*}$ . This suggests that if  $\pmb{f}^t$  is far from  $\pmb{f}^{*}$ , then matrix  $A^t$  can potentially decrease  $\mathcal{L}$  significantly at the best case while not causing any increase at the worst case.
Theorem 13. Suppose the communication in the  $t$ -th iteration of multiple learners is denoted by the matrix  $A^t$  and returns  $\pmb{f}_{A^t}^{t+1} \in \mathcal{H}^d$ , for both RFT and GFT we have:
$$
\mathbb {E} _ {\boldsymbol {x} \sim [ \mathbb {P} _ {i} (x _ {i}) ] ^ {d}} \left[ \mathcal {L} \left(\boldsymbol {f} _ {A ^ {t}} ^ {t + 1}\right) - \mathcal {L} (\boldsymbol {f} ^ {t}) \right] \leq \mathbb {E} _ {\boldsymbol {x} \sim [ \mathbb {P} _ {i} (x _ {i}) ] ^ {d}} \left[ \mathcal {L} \left(\boldsymbol {f} _ {A ^ {t}} ^ {t + 1}\right) - \mathcal {L} \left(A ^ {t} \boldsymbol {f} ^ {t}\right) \right] \leq 0. \tag {17}
$$
(a) Single-learner teaching.
(b) Vanilla MINT.
Figure 4: Visualization of  $\pmb{f}^{t}$  taught by GFT. Starting from a random initialization, the communicated multi-learner GFT help multiple learners learn a more clear image than the vanilla one followed by single-learner one.
(c) Communicated MINT.
Proof of Theorem 13 is in Appendix B. This shows that the addition of communication has led to an improvement in model updates, which is evident from the larger loss discrepancy between  $f_{A^t}^{t+1}$  and  $f^t$  compared to the difference observed between  $f_{A^t}^{t+1}$  and  $A^t f^t$ .
Testing the teaching of a multi-learner (vector-valued) target model, MINT presents more satisfactory performance than repeatedly carrying out the single-learner teaching, which is consistent with our theoretical findings. Detailed configurations and supplementary experiments are given in the Appendix C.
MINT in gray scale. A grayscale figure can be viewed as a 3D surface where the  $z$  axis corresponds to the level of gray, while the  $x, y$  axes depict the placement of pixels [73]. We consider two scenarios: one involves the simultaneous teaching of a tiger and a cheetah figure, while the other focuses on the teaching of a lion. After comparing (a) and (b) in Figure 2, we see that when teaching two target functions by GFT simultaneously, the vanilla MINT requires almost half the number of cost iterations compared to single-learner teaching, which is also evident from the loss plot shown in Figure 3 (a). By comparing (c) and (d) in Figure 2, we can observe that dividing a single-learner target figure into smaller pieces and recasting them into MINT can significantly improve the efficiency, which is also demonstrated by the loss plot in Figure 3 (b).
MINT in three (RGB) channels. To further demonstrate the benefits of communication, we examine with a lion image with three channels in RGB format. The loss plot in Figure 3 (c) reveals that the most efficient teaching is the communicated MINT for both RFT and GFT. The vanilla MINT and single-learner teaching follow in order of decreasing efficiency. Furthermore, as anticipated, the multi-learner GFT proves to be more efficient compared to RFT. One intriguing observation is that the communicated MINT leads to a significant reduction in multi-learner loss at the outset, which aligns with our theoretical findings in Lemma 12 and confirms the validity of Prop.11 that  $A^t$  could eventually become an identity matrix after numerous iterations. Figure 4 compares the specific learnt  $f^t$  for three versions of GFT during each iteration, wherein we observe that MINT consistently outperforms the single-learner one, and the learnt image under the communicated MINT is more clear compared to that of the vanilla one. To be more persuasive, we also offer detailed and additional
Our source code is available at https://github.com/chen2hang/MINT_NonparametricTeaching.
experiments in Appendix, including channel-wise visualization of specific  $\pmb{f}^{t}$  (Figure 7), RFT-taught  $\pmb{f}^{t}$  (Figure 8-9) and teaching multiple learners with a particular initialization of  $\pmb{f}^{0}$  (Figure 11-10), which includes an extreme case that only one-time communication is sufficient to help multiple learners learn  $\pmb{f}^{*}$  (Figure 15).
6 Concluding Remarks and Future Work
In this paper, we seek to address a practical limitation of current nonparametric iterative machine teaching by enabling the teaching of multi-learner (vector-valued) target models. This generalization of teaching ability involves generalizing the model space from space of scalar-valued functions to that of vector-valued functions. In order to address multi-learner nonparametric teaching, we start by analyzing a vanilla MINT where the teacher picks examples based on a vector-valued target function such that multiple learners can learn its components simultaneously. Additionally, we consider the communicated MINT (i.e., multiple learners are allowed to carry out linear combination on current learnt functions) for further exploration. Through both theoretical analysis and empirical evidence, we demonstrate that the communicated MINT is more efficient than the vanilla MINT.
Moving forward, it could be interesting to explore other practical aspects related to nonparametric teaching. This will involve a deeper theoretical understanding and the development of more efficient teaching algorithms. Besides, it would be intriguing to establish connections between MINT and multi-output neural networks, which can further enhance its practical applications such as knowledge distillation. Moreover, generating teaching examples with a surrogate objective that does not need a target model (e.g., black-box teaching) is also an important direction (e.g., [17, 72]). More generally, (iterative) machine teaching is intrinsically connected to the recent popular data-centric AI. Understanding data-centric learning (e.g., text prompting, data augmentation, data distillation) may require a deeper understanding towards (iterative) machine teaching.
This work was supported in part by National Natural Science Foundation of China (Grant Number: 62206108), in part by Maritime AI Research Programme (SMI-2022-MTP-06) and AI Singapore OTTC Grant (AISG2-TC-2022-006), and in part by the Research Grants Council of the Hong Kong Special Administrative Region (Grant 16200021).
[1] Robert A Adams and John JF Fournier. Sobolev spaces. Elsevier, 2003. 3
[2] Scott Alfeld, Xiaojin Zhu, and Paul Barford. Data poisoning attacks against autoregressive models. In AAAI, 2016. 1
[3] Scott Alfeld, Xiaojin Zhu, and Paul Barford. Explicit defense actions against test-set attacks. In AAAI, 2017. 1
[4] Maurizio A Alvarez, Lorenzo Rosasco, Neil D Lawrence, et al. Kernels for vector-valued functions: A review. Foundations and Trends® in Machine Learning, 4(3):195-266, 2012. 2, 3
[5] Michael Arbel, Anna Korba, Adil Salim, and Arthur Gretton. Maximum mean discrepancy gradient flow. In NeurIPS, 2019. 3, 4
[6] Axel D Becke. Density-functional exchange-energy approximation with correct asymptotic behavior. Physical review A, 38(6):3098, 1988. 3
[7] Jon Louis Bentley. Multidimensional divide-and-conquer. Communications of the ACM, 23(4):214-229, 1980. 2
[8] Dimitri P Bertsekas and John N Tsitsiklis. Introduction to probability vol. 1. 2002. 26
[9] Andrea Caponnetto, Charles A Micchelli, Massimiliano Pontil, and Yiming Ying. Universal multi-task kernels. The Journal of Machine Learning Research, 9:1615-1646, 2008. 3
[10] Claudio Carmeli, Ernesto De Vito, and Alessandro Toigo. Vector valued reproducing kernel hilbert spaces of integrable functions and mercer theorem. Analysis and Applications, 4(04):377-408, 2006. 2, 3
[83] Xiaojin Zhu, Adish Singla, Sandra Zilles, and Anna N Rafferty. An overview of machine teaching. arXiv preprint arXiv:1801.05927, 2018. 1, 3
[84] R Zoppoli, M Sanguineti, and Thomas Parisini. Approximating networks and extended ritz method for the solution of functional optimization problems. Journal of Optimization Theory and Applications, 112(2):403-440, 2002. 3
This paper is to address a practical limitation of recently introduced Nonparametric Iterative Machine Teaching from the theoretical aspect, which is to enable multi-learner teaching. We also provide empirical evidence to demonstrate that multi-learner nonparametric teaching is effective in addressing such a limitation. Although we hope that the outcomes of this study will enlighten the theoretical community, we do not expect any immediate effects on society from this work.
Pseudo code for multi-learner RFT and GFT
The pseudo code for RFT and GFT in the vanilla and communicated MINT is given as following:
Algorithm 1 Greedy (Random) Functional Teaching for the Communicated (Vanilla) MINT
Input: Target  $\pmb{f}^{*} \in \mathcal{H}^{d}$ , initial  $\pmb{f}^{0} \in \mathcal{H}^{d}$ , small constants  $\epsilon, \epsilon_{0} > 0$  and maximal iteration numbers  $T, T_{0}$ .
Set  $f^t \gets f^0, t = 0$ .
while  $t\leq T$  and  $\| f^t -f^*\|_{\mathcal{H}^d}\geq \epsilon \mathrm{d}\boldsymbol {0}$
The teacher is not only to construct the communication matrix but alos to select the teaching set for multiple learners:
// Construction of the communication matrix.
Initialize a two-layer perceptron with a linear layer, i.e., initialize the communication matrix with an identity one ( $A = I_d$ ), and set  $t_0 = 0$ .
while  $t_0\leq \bar{T}_0$  and  $\| A\pmb {f}^t -\pmb {f}^*\|_{\mathcal{H}^d}\geq \epsilon_0$  do
Train the linear weight  $(A)$  of this two-layer perceptron, such that  $\| A\pmb{f}^t -\pmb {f}^*\|_{\mathcal{H}^d}$  decreases. end
/\*For the Vanilla MINT, omit the above procedure of solving for the communication matrix, just need to set  $A = I_{d}$  \*/
// Selection of the teaching set.
Initialize the teaching set  $\mathcal{D} = \emptyset$
Pick  $x_{i}^{t^{*}} \in \mathcal{X}_{i}, i \in \bar{\mathbb{N}}_{d}$  with the maximal difference between  $f_{i}^{t}$  and  $f_{i}^{*}$  for multiple learners simultaneously:
$$
x_{i}^{t^{*}} = \operatorname *{arg  max}_{x_{i}^{t}\in \mathcal{X}_{i}}\left|f_{i}^{t}(x_{i}^{t}) - f_{i}^{*}(x_{i}^{t})\right|;
$$
/\* For random functional teaching, Pick  $x_{i}^{t^{*}}\in \mathcal{X}_{i},i\in \mathbb{N}_{d}$  randomly and concurrently.
Add  $\left(x_{i}^{t^{*}},y_{i}^{t^{*}} = f_{i}^{*}\left(x_{i}^{t^{*}}\right)\right)$  into the teaching set  $\pmb{D}$ .
Provide  $A$  and  $\mathcal{D}$  to multiple learners.